Question: Simplify the following expression and state the condition under which the simplification is valid. $r = \dfrac{-7y^2 - 7y + 84}{9y^2 + 27y - 36}$
Explanation: First factor out the greatest common factors in the numerator and in the denominator. $ r = \dfrac {-7(y^2 + y - 12)} {9(y^2 + 3y - 4)} $ $ r = -\dfrac{7}{9} \cdot \dfrac{y^2 + y - 12}{y^2 + 3y - 4} $ Next factor the numerator and denominator. $ r = - \dfrac{7}{9} \cdot \dfrac{(y + 4)(y - 3)}{(y + 4)(y - 1)}$ Assuming $y \neq -4$ , we can cancel the $y + 4$ $ r = - \dfrac{7}{9} \cdot \dfrac{y - 3}{y - 1}$ Therefore: $ r = \dfrac{ -7(y - 3)}{ 9(y - 1)}$, $y \neq -4$